How To Calculate The Radius Of The Earth

Here is a simple procedure to calculate the radius of the Earth:

• Find someone over 500 km due north or due south of your position that will measure the length of the shadow of a vertical stick at some agreed upon date and time.
• On the same day, simultaneously measure the length of the shadow of a vertical stick at your location.
• Calculate R, the radius of the Earth, using R = s / (σ - φ) where:
  • s is the arc length between both vertical sticks.
  • σ = arctan(b / a)
  • φ = arctan(d / c)
  • a is the height of their vertical stick.
  • b is the length of the shadow of their vertical stick,
  • c is the height of your vertical stick.
  • d is the length of the shadow of your vertical stick.

Finding someone due north or due south of your position and s will likely require a map. σ = arctan(b / a) because tan σ = b / a. φ = arctan(d / c) because tan φ = d / c. R = s / (σ - φ) because σ - φ = s / R in radians. The two appearances of σ and φ in the diagram are due to the congruency of alternate interior angles formed from parallel light rays from the sun and the transversals collinear with the vertical sticks.

Eratosthenes performed a similar calculation about 2200 years ago!